KAM and Melnikov theory describing island chains in plasmas

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KAM and Melnikov theory

describing island chains in plasmas

Thesis submitted in partial fulfillment of the requirements for the degree of

BACHELOR OFSCIENCE in

MATHEMATICS ANDPHYSICS

Author : Joost Opschoor

Student ID : 1221809

Supervisors physics : dr. Jan Willem Dalhuisen prof. dr. Dirk Bouwmeester Supervisor mathematics : prof. dr. Arjen Doelman

Leiden University

Leiden Institute of Physics & Mathematical Institute Leiden, The Netherlands, July 15, 2016

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KAM and Melnikov theory

describing island chains in plasmas

Joost Opschoor

Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

July 15, 2016

Abstract

In this thesis toroidal plasmas are described by a Hamiltonian system parameterising the magnetic field lines. Application of KAM and Mel-nikov theory explains the occurance of chaotic island chains bounded by invariant manifolds. In plasma simulations and experiments island chains have shown to be rather stable. This thesis provides a theoretical but nonrigorous argument for this stability and proposes the description of magnetohydrodynamics as a Hamiltonian field theory as a method to rigorously study the stability. Hypotheses are given for a generalisation to nontoroidal structures.

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Introduction

Since the 1950’s toroidal plasmas have been of great importance due to their application in nuclear fusion reactors, for which it is needed create a stable plasma under extreme conditions. An ideal way to achieve this goal is to use a plasma that is stable due to its own structure. One approach to obtain self-stability is to use a plasma with a high magnetic helicity, which corresponds to magnetic field line linking. This creates special interest in a plasma with the magnetic field derived from the Hopf map of which all field lines are circles linked with all other field lines exactly once. In addition to that the field lines fill tori, such that the whole space is foliated by tori.

This bachelor project has been performed in the Dirk Bouwmeester group at Leiden University, which does experimental and theoretical research on self-organising knot-ted magnetic structures, including structures similar to the one with the Hopf field as magnetic field. Results from their simulations of nonideal plasmas ([1]) show toroidal structures of which the magnetic field is similar to structures described by KAM and Melnikov theory for small perturbations of completely integrable Hamiltonian systems. A completely integrable Hamiltonian system has toroidal invariant manifolds in phase space, each orbit lies on such a torus. This is similar to the tori filled with field lines in the magnetic field derived from the Hopf map. Perturbations to the Hamiltonian system lead to chaotic island chains described by Melnikov theory. In plasma physics magnetic island chains with a similar structure are observed. In fusion reactors magnetic islands can blow up and explode onto the wall or distort the plasma stability in other ways, which makes their behaviour very important. In Reference [1] the dynamics of mag-netic island chains is different: they shrink and move to the center of the plasma where they disappear. Despite the chaotic island chains KAM theory proves the preservation of invariant manifolds in phase space in small perturbations of completely integrable Hamiltonian systems. Toroidal plasmas also contain tori completely filled with field lines that form barriers between chaotic island chains.

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Aim of this project

Despite the similarity described above plasma physics and Hamiltonian systems are two completely different concepts, the correspondence between the two is nontrivial. The aim of this thesis is to make this correspondence exact and to show how KAM theory and the Melnikov theory can be used to explain observations in plasma physics, namely the preservation of impenetrable manifolds and the existence of chaotic island chains. Besides that an argument will be given for the relatively stable behaviour of these plasma structures. Structures with similar properties but a nontoroidal shape have been observed ([2]), hypotheses will be given about their behaviour.

Two correspondences between plasma physics and Hamiltonian systems

This thesis is based on two correspondences between plasma physics and Hamiltonian systems. The first one is based on the idea that the magnetic field of a plasma represents the structure of the plasma. The magnetic field can be related to a Hamiltonian sys-tem through its divergencelessness, the correspondence between divergenceless vector fields and Hamiltonian systems has been shown in general in Reference [3]. For the second correspondence a set of partial differential equations that describes the plasma is written as an infinite-dimensional Hamiltonian dynamical system to which Hamilto-nian field theory can be applied. This shows the HamiltoHamilto-nian nature of that particular model for plasma physics, ideal magnetohydrodynamics. The infinite-dimensional dy-namical system is mathematically far more complex than the finite-dimensional Hamil-tonian system of the first approach, therefore the first will be used throughout this thesis while the second is proposed as a possible direction for further research.

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Overview of this thesis

This thesis roughly consists of two parts: Chapters 1 - 6 describe theories and models for plasma physics and Hamiltonian systems, which are used to explain plasma physcis observations in Chapters 7 - 8. An overview per chapter is given below.

Chapter 0 gives preliminary remarks about smoothness and the definition of a manifold. Paragraphs 1.1,1.2 introduce magnetohydrodynamics (MHD), a model for plasma physics. Paragraphs 1.3,1.4 describe an ideal stationary plasma situation with a mag-netic field derived from the Hopf map and generalisations of that field. It will be used as the primary examples of ideal toroidal plasmas.

Chapter 2 introduces Hamiltonian systems and related concepts and methods, includ-ing complete integrability, invariant manifolds, action-angle variables, the reduction of Hamiltonian systems with an invariant and Poincar´e maps.

Chapter 3 makes the first connection between dynamical systems and vector fields. It describes a vector field as a dynamical system using functions that parameterise field lines of the vector field, which are orbits of the dynamical system. This correspondence can be used to study the behaviour of zeroes of the vector field and field lines converg-ing to or divergconverg-ing from a zero.

Chapter 4 uses a similar method to describe divergenceless vector fields. The diver-gencelessness is used to give the dynamical system a Hamiltonian structure. Intuitively the volume preservation of flux tubes moving along vector field lines corresponds to the volume preservation in Hamiltonian phase space. This chapter discusses one very explicit method to describe this correspondence, which makes it useful for applications. Other correspondences are mentioned in the introduction to the chapter. In Paragraph 4.4 this method is applied to the Sagdeev fields studied in Paragraph 1.4, giving a Hamiltonian system with a toroidal motion in phase space, which can be generalised for other toroidal fields. Paragraph 4.5 shows how symmetry can be used to describe vector fields with a nontoroidal structure in terms of a toroidal Hamiltonian system.

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In Chapter 5 the Hamiltonian system derived in Paragraph 4.4 is generalised to Hamil-tonian systems containing island chains. These systems contain both idealised and re-alistic models for island chains and they are used to describe the basic island chain properties. Paragraph 5.3 describes singularities that come into play if the method of Chapter 4 is used to relate these generalised Hamiltonian systems to divergenceless vector fields, a correction for singularities is given as well.

Chapter 6 covers Hamiltonian system theories that describe the structure of island chains present in the models of Chapter 5. It gives two variants of a KAM-theorem describing the preservation of invariant tori for “small enough” perturbations of com-pletely integrable Hamiltonian systems and describes the emergence of chaotic island chains in systems with such perturbations. The chaotic structure is stated with results from Melnikov theory. For larger perturbations the chaotic structure is preserved and an estimative but nonrigorous method shows that often many invariant tori are preserved. Paragraphs 7.1,7.2 bring the theory of Chapters 4 - 6 together to explain the rather sta-ble structure of chaotic island chains bounded by impenetrasta-ble manifolds observed in time dependent plasmas. Paragraph 7.3 describes the time dependent ideal MHD equa-tions of Paragraph 1.1 as an infinite-dimensional dynamical system with the form of a Hamiltonian field theory, which can be used to study the stability of the predescribed time-dependent structures.

The outlook of Chapter 8 describes the current view on some recent and partly unpub-lished observations in simulations. This view can be used as hypotheses for further research. It includes the relation between the shape of a nontoroidal slowly decaying structure and the symmetry of the initial condition of the simulation. Besides that the (im)possibility to describe those structures by a Hamiltonian system with a toroidal structure is discussed as well as the motion of island chains due to different profiles of the rotational transform as observed in the simulations of Reference [1].

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Chapter 0 Preliminary remarks

Smoothness

In this thesis all structures are smooth, unless stated otherwise. As an example the term “manifold” is used for a differentiable manifold, defined below. Familiarity with differentiable manifolds is not needed to understand this thesis, it is enough to keep in mind the intuitive idea of a set that is locally diffeomorphic toRd.

Definition 1. A set M ⊂Rn _{is a smooth manifold in}_Rn _{if for each p} _∈ _{M there exists a an}

open set U ∈ Rd and a diffeomorphism ϕ : U → M such that 0 7→ p. The map ϕ is called a parameterisation and d is the dimension of the manifold.

Manifolds defined as above are sometimes referred to as smooth manifolds inRn without boundary.

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Chapter 1 Magnetohydrodynamics

Plasma is a phase of matter where nearly all atoms are ionised, a plasma consists of ions and separate electrons. As all particles are charged, they interact with electromagnetic fields. A model for the charged particles interacting with electromagnetic fields can be used as the description of a plasma, but for practical purposes simplified models have been constructed, one of which is magnetohydrodynamics (MHD). It treats the plasma as a fluid interacting with electromagnetic fields. In principle it is a combination of Maxwell’s equations and fluid dynamics, simplified for a specific regime of plasma parameters. This approximation is a huge simplification, but it turns out to describe most properties of interest very well, even outside the regimes for which it was set up in the first place. This is also the case for the plasma structures that are the topic of this thesis, which justifies the use of MHD.

The following paragraphs give the equations that define ideal and resistive magneto-hydrodynamics: IMHD in Paragraph 1.1 and RMHD in Paragraph 1.2, for reference see [4]. The specific plasma conditions used by Kamchatnov and Sagdeev are discussed in Paragraph 1.3 while the fields they studied are described in Paragraph 1.4. Those plasma situations are important because they form the basis for explicit plasma models studied in Chapter 5.

1.1 Ideal magnetohydrodynamics (IMHD)

This subsection gives the equations that define ideal magnetohydrodynamics. IMHD is a form of magnetohydrodynamics in which there is no electromagnetic resistivity or viscosity and in which the conductance is perfect.

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4 Magnetohydrodynamics

Definition of quantities:

τ Time

ρ Mass density

~v Velocity field field

p Scalar hydrodynamic pressure

~_B _{Magnetic field}

γ Poisson constant, ratio of specific heats

∂ρ ∂τ + ∇ · (ρ~v) = 0 (1.1) ∂~B ∂τ − ∇ × (~v× ~B) = 0 (1.2) ρ ∂~v ∂τ + ~v· ∇~v + ∇p− 1 µ0 (∇ × ~B) × ~B=0 (1.3) ∂ p ∂τ+ ~v· ∇p +γ p∇ · ~v =0 (1.4) ∇ · ~B=0 (1.5)

Equation 1.1 describes the conservation of mass, and 1.2 is Gauss’ law simplified using the absense of electromagnetic resistivity: ~E+ ~v× ~B = 0. The force balance is given by 1.3 and 1.4 is the energy equation. Although 1.5 should be fulfilled at any time, it is enough to impose it as an initial condition, because Equation 1.2 implies that it remains fulfilled.

1.2 Resistive magnetohydrodynamics (RMHD)

This subsection describes the equations that define RMHD. The nonidealities which are introduced with respect to IMHD are resistivity and viscosity. Other nonidealities can be introduced as well, they are not included here. The aim of this subsection is to give an example of a nonideal model, not to give a complete overview of nonidealities.

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1.3 Conditions used by Kamchatnov and Sagdeev 5

Definition of quantities:

τ Time

ρ Mass density

~v Velocity field

p Hydrodynamic pressure (scalar field)

~_B _{Magnetic field} ~_E _{Electric field}

γ Poisson constant, ratio of specific heats

~_j _{Electrical current density} ~_F_visc _{Viscosity force}

η Electromagnetic resistivity ∂ρ ∂τ + ∇ · (ρ~v) =0 (1.6) ∂~B ∂τ + ∇ × ~E =0 (1.7) ρ ∂~v ∂τ + ~v· ∇~v + ∇p−~j× ~B =0 (1.8) ∂ p ∂τ + ~v· ∇p +γ p∇ · ~v= ~Fvisc (1.9) ∇ · ~B=0 (1.10) ~_E_{+ ~}_v_{× ~}_B₌_η~_j _(1.11) ~_j ₌ 1 µ0 ∇ × ~B (1.12)

Viscosity has been added to Equation 1.9, while electromagnetic resistivity is described by Equation 1.11. IMHD corresponds to~Fvisc =0, η =0.

1.3 Conditions used by Kamchatnov and Sagdeev

In Reference [5] Kamchatnov studied a stationary solution of the IMHD equations, which means that the time derivatives of ~B,~v, p, ρ vanish. Besides that the fluid was assumed to be incompressible (∇ · ~v = 0) and the mass density to be homogenious (∇ρ =0).

The incompressibility implies that the ratio of specific heats, γ = C_Cp

v, is infinitely large.

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by neglecting the first term. The resulting equation is equal to the assumption of an incompressible fluid. There has been a discussion about the validity of this reduction, in this thesis it will just be assumed.

After the application of stationarity and incompressibility the IMHD equations are as given below, homogeneity has not been expressed in these equations.

~v· ∇ρ=0 (1.13) ∇ × (~v× ~B) =0 (1.14) ρ~v· ∇~v+ ∇p− 1 µ0 (∇ × ~B) × ~B =0 (1.15) ∇ · ~v =0 (1.16) ∇ · ~B=0 (1.17)

Homogeneity solves Equation 1.13. Equations 1.14,1.16,1.17 have been solved by taking

~v k ~B for a divergenceless vector field ~B. This assumption is supported by the fact that charged particles will gyrate by the magnetic field, which drastically restricts the motion perpendicular to the magnetic field, particles (nearly) move parallel to magnetic field lines.

The only equation that has not been solved is 1.15, which is equivalent to:

~v· ∇~v− 1 µ0ρ(~B· ∇~B) + 1 ρ∇ p+ B 2 2µ0 =0 (1.18)

This equation has been solved by taking~v to be the Alfv´en speed and by taking a specific pressure profile ~v = ±√~B µ0ρ (1.19) p = p∞− B 2 2µ0 (1.20)

for a constant p∞. The constant p∞ is equal to the pressure at infinity if it is assumed

that~B converges to 0 at infinity. Note that p is determined by~B up to the constant p∞.

With this construction Kamchatnov found a specific solution of equations 1.14 - 1.17 based on a divergenceless field ~B - the Hopf field. The plasma configuration is fully described by the magnetic field (apart from the density and p∞, which are free

parame-ters).

The assumptions of equations 1.19,1.20 can be made for every divergenceless vector field~B, which gives a stationary solution of the IMHD equations for each such vector field~B. This supports the idea that a divergenceless field describes a plasma situation,

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1.4 Magnetic field of the IMHD solutions studied by Kamchatnov and Sagdeev 7

in the rest of this thesis it will be assumed that a divergenceless vector field represents the structure of the plasma situation.

The Hopf field used by Kamchatnov and its generalisations by Sagdeev described in the next paragraph will be used as an example on which more general models for plasma physics will be based. Besides the solution of IMHD studied by Kamchatnov and Sagdeev there are other similar stationary solutions of IMHD, for example the so-lutions with a compressive fluid studied in Reference [6]. They can be used as examples instead of those by Kamchatnov and Sagdeev.

This paragraph has shown that under certain assumptions the structure of a stationary solution of IMHD is completely described by the magnetic field - a divergenceless vector field defined onR3. Examples of such fields will be described in the next paragraph.

1.4 Magnetic field of the IMHD solutions studied by

Kamchatnov and Sagdeev

Formulas for Kamchatnov and Sagdeev fields

Kamchatnov studied an IMHD solution of which the magnetic field is derived from the Hopf map ([5]). In terms of cartesian coordinates it is given below.

~_B ₌ 4 √ a π(1+x2+y2+z2)3    2y−2xz −2x−2yz (x2+y2−z2−1)    (1.21)

In this formula a is a parameter that determines the magnetic field strength at the origin. It follows from its construction that all field lines except the field line filling the z-axis are circles which are linked exactly once with each other and with the z-axis (Figure 1.1). The linked field lines give nonzero magnetic helicity Hm =

R _~

A· ~Bd3x, which a con-served quantity in IMHD.

In Reference [7] Sagdeev described a generalisation of the field studied by Kamchatnov, with the following magnetic field:

~ B = 4 √ a π(1+x2+y2+z2)3    2ω2y−2ω1xz −2ω2x−2ω1yz ω1(x2+y2−z2−1)    (1.22)

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Figure 1.1:Field lines of the divergenceless vector field derived from the Hopf map.

The followig equation gives a corresponding vector potential (a vector fieldA such that~

~_B_{= ∇ × ~}_A): ~ A= √ a π(1+x2+y2+z2)2    2ω1y−2ω2xz −2ω1x−2ω2yz ω2(x2+y2−z2−1)    (1.23)

Note that the Sagdeev field with ω1 =1 = ω2is equal to the Hopf field used by

Kam-chatnov.

This thesis will use a slightly modified form of these formulas, which simplifies later expressions. a=ω−₂2will be used and the gradient1

ω1 2πω2     −_x₂₊y_y₂ x x2₊_y2 0    

1_{This expression is the gradient} ω1∇tr

2πω2 where t

r_{is the toroidal coordinate c}

1as defined in appendix B.

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1.4 Magnetic field of the IMHD solutions studied by Kamchatnov and Sagdeev 9

will be added to the vector potential after filling in a = ω−₂2. These modifications can be used without loss of generality for ω2 6= 0. If ω2 = 0 the unmodified version can be

used.

The modifications give the following expressions:

~_B ₌ 4 π(1+x2+y2+z2)3    2y−2ω1 ω2xz −2x−2ω1 ω2yz ω1 ω2(x 2₊_y2₋_z2₋₁₎    (1.24) ~ A = 1 π(1+x2+y2+z2)2    2ω1 ω2y−2xz −2ω1 ω2x−2yz (x2+y2−z2−1)   + ω1 2πω2     −_x2₊y_y2 x x2₊_y2 0     (1.25)

Field line structures for the Sagdeev fields For ω1

ω2 ∈ Q the field lines of the Sagdeev fields are closed, they are torus knots, while

for ω1

ω2 ∈/Q field lines densely fill a torus (Figure 1.2).

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For the Sagdeev fields all field lines lie on tori, except for a field line on the unit circle and a field line on the z-axis. Therefore the field can best be described in a toroidal coordinate system, as discussed in Paragraph 4.4. Appendix B describes a suitable co-ordinate system (q1, ρ, q2) in which the vector potential of the Sagdeev fields is given

by Equation 4.26, which is copied below. The notation∇q1,∇q2is used for basis vector

fields, it can be interpreted as a gradient with respect to another coordinate system.

~ A = sech 2 ρ 2π ∇q2− ω1sech2ρ 2πω2 ∇q1 10

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Chapter 2 Concepts and methods for Hamiltonian

systems

This section gives some definitions and concepts for Hamiltonian dynamical systems. A completely integrable Hamiltonian system is the aim of the construction in Chapter 4 and the starting point for the theory of Chapter 6.

Many concepts discussed in this chapter are widely used, those concepts will not be discussed thoroughly.

2.1 Definitions of Hamiltonian systems, separability and

integrability

Definition 2. AHamiltonian system with n degrees of freedom is a dynamical system of 2n functions depending on t: p₁t(t), . . . , pt_n(t), q₁t(t), . . . , qt_n(t)satisfying Hamilton’s equations: the following system of 2n ordinary differential equations.

dpt_i(t) dt = − ∂H ∂qi (pt(t), qt(t), t); dq t i(t) dt = ∂H ∂ pi (pt(t), qt(t), t) (2.1) The differential equations depend on the function H(p1, . . . , pn, q1, . . . , qn, t), which is called the

Hamiltonian. The variable t is called Hamiltonian time variable. The functions pt_i(t), qt_i(t) : 1≤i ≤n solving 2.1 together form an orbit of the Hamiltonian system. Orbits will be denoted as follows:

pt(t) .._{= (}_p₁t₍_t₎_{, . . . , p}t_n₍_t₎₎

qt(t) .._{= (}_qt

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12 Concepts and methods for Hamiltonian systems

qiare called position variables, while pi are generalised momenta. For a fixed i the variables

pi, qiare a pair of conjugate variables.

On orbits the Hamiltonian can be seen as a function of t:

Ht(t) ..₌_H₍_pt₍_t₎_{, q}t₍_t₎_{, t}₎ _(2.2)

Hamilton’s equations imply that Ht only depends on t through its partial derivative with respect to time:

dHt(t) dt = n

∑

i=1 ∂H ∂ pi (pt(t), qt(t), t)dp t i(t) dt + ∂H ∂qi (pt(t), qt(t), t)dq t i(t) dt + ∂H ∂t (p t₍_t₎_{, q}t₍_t₎_{, t}₎ = ∂H ∂t (p t₍_t₎ , qt(t), t) (2.3)

Definition 3. If the Hamiltonian does not directly depend on time, i.e. ∂H

∂t = 0, the system is

called autonomous.

Every nonautonomous Hamiltonian system with n−1 degrees of freedom (having n−1 pi’s and qi’s) can be transformed into an autonomous n degree of freedom Hamiltonian

system (having n pi’s and qi’s). An example of this procedure is explained on page 258

of Reference [8] and in essence it is the inverse transformation of the reduction described in Paragraph 2.4.

Such systems are said to have n−1

2 degrees of freedom, independent of the number of

pi, qi’s in the description. In this thesis a different convention will be used: the different

notations will be denoted as the n−1 degree of freedom system and the n degree of freedom system.

Separability

Hamiltonian systems for which pi, qiand pj, qjdo not interact for i 6= j are called

separa-ble. The Hamiltonian can be written in the form

H(p, q, t) = F1(p1, q1, t) +. . .+Fn(pn, qn, t) (2.4)

for n functions Fi. Separable Hamiltonian systems can be seen as n independent one

de-gree of freedom Hamiltonian systems with Hamiltonians Fi. This means that separable

systems have a relatively simple structure.

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2.2 Geometric representation of a Hamiltonian system 13

Invariants and integrability

Another important property of Hamiltonian systems is the existence of invariants: Definition 4([9] page 333). A function f(p, q)is an invariant if

d f(pt(t), qt(t))

dt =0 (2.5)

for all orbits of the Hamiltonian system. Invariants are assumed to have nonvanishing deriva-tives.

Invariants are important because they constrain the orbits of the Hamiltonian system to manifolds on which the invariants are constant. Therefore it “reduces the dimension of the system”, just as the dynamics of separable dynamical systems is reduced to one-dimensional Hamiltonian systems.

The existence of invariants is often called integrability. A notion of integrability that is important for the KAM theorems in Paragraph 6.1 is defined below.

Definition 5([9] page 368). A Hamiltonian system is completely integrable in the sense of Liouville if there exist n invariants which are linearly independent almost everywhere and for which the following equation holds.

0= {fj, fk}..= n

∑

i=1 ∂ fj ∂ pi ∂ fk ∂qi −∂ fk ∂ pi ∂ fj ∂qi (2.6) Two invariants satisfying 2.6 are said to be in involution.

For autonomous Hamiltonian systems the Hamiltonian is an invariant. Therefore all autonomous systems with one degree of freedom are completely integrable.

2.2 Geometric representation of a Hamiltonian system

Just as all other dynamical systems Hamiltonian systems can be represented geometri-cally by vector fields, which is one of the main ideas of the mathematical field of dy-namical systems. It allows the use of geometric arguments in the study of differential equations. This paragraph is a short overview of this correspondence. Besides that the important concept of invariant manifolds will be introduced.

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A vector field on phase space

The domain of the vector field representing the Hamiltonian system is a 2n-dimensional manifold M that contains all values (p, q) that orbits of the dynamical system can as-sume. It is called phase space and has coordinate fields p1, . . . , pn, q1, . . . , qn.

An autonomous Hamiltonian system can be seen as a family of maps{ϕt}t∈R, ϕt : M →

Msuch that∀s, t ∈R : ϕt◦ϕs = ϕt+s. This family of maps is called the flow.

The orbits pt, qt of such a system can also be defined as the orbit of a point in phase space(pt(t0), qt(t0))under the flow:

(pt(t), qt(t)) = ϕt−t0(p

t₍_t

0), qt(t0)) (2.7)

Hamilton’s equations define a vector field on phase space, namely the t-derivative of the flow ([9] page 108):

dϕt dt (p, q) = dpt₁ dt (p, q), . . . , dpt_n dt (p, q), qt₁ dt(p, q), . . . , dqt_n dt (p, q) (2.8)

Orbits of the dynamical system correspond to field lines of this vector field. The field lines can be parameterised by t, just as the orbits of the dynamical system. Expressions like dpt1

dt correspond to partial derivatives of the Hamiltonian, which do not depend on

t, which means that the vector field described above does not depend on t.

Invariant manifolds

Invariants can also be described geometrically, in terms of invariant manifolds and foli-ations.

Definition 6. A submanifold of phase space is an invariant manifold if for all for all t ∈ R

and all points(p0, q0)in the manifold the image ϕt(p0, q0)under the flow lies in that manifold.

The interpretation of invariant manifolds in terms of the vector field on phase space is that for all points in an invariant manifold the field line through that point completely lies in the manifold. This implies that field lines of the vector field cannot cross invariant manifolds. That is what makes the existence of invariant manifolds important for the description of plasma physics: invariant manifolds act as boundaries for the dynamics. For an invariant f(p, q) its level sets {f = c} are 2n−1-dimensional invariant mani-folds. The collection{f =c}_c∈Rfoliates phase space:

Definition 7. A family of pairwise disjoint submanifolds of phase spacefoliates phase space if the union of these submanifolds equals phase space.

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2.3 Action-angle variables 15

Completely integrable systems have n independent invariants, corresponding to n fam-ilies of 2n−1-dimensional invariant manifolds such that each family foliates phase space. Intersections of invariant manifolds are invariant manifolds themselves. This leads to the following proposition:

Proposition 1([9] page 368 Theorem 9.19). The phase space of a completely integrable Hamil-tonian system is foliated by n-dimensional invariant manifolds.

If those manifolds are compact and connected, they are diffeomorphic to n-tori, S1×. . .×S1

| {z }

n times

. This structure is similar to the structure of the Sagdeev fields discussed in paragraph 1.4, where field lines lie on two-dimensional tori. The intuitive similarity between the two is made exact in chapter 4.

2.3 Action-angle variables

Completely integrable autonomous Hamiltonian systems with compact and connected invariant manifolds have a very rich structure. In order to stress this structure and to simplify its use such systems are often transformed to “action-angle variables” de-fined below. Transformations should preserve the Hamiltonian structure, which is why canonical coordinate transformations are used. In order to ensure that the coordinate transformation is canonical it can be constructed using a generating function, a definition and an explanation can be found in on page 126 of Reference [10]. If the generating function solves the stationary Hamilton-Jacobi equation, a partial differential equation, then the resulting coordinate system is given in action-angle variables ([10] page 127). That shows the difficulty of constructing action-angle variables.

Definition 8. A pair of conjugate variables pi, qiare action-angle variables if the Hamiltonian

does not depend directly on qi:

∂H ∂qi =0⇒ dp t i dt =0 ; ∂H ∂ pi = dq t i dt

In this case piare called action-variables and qiare called angle-variables.

This definition implies that action-variables pi are independent invariants which are in

involution. If all n pairs of conjugate variables are given in action-angle form and if the manifolds of constant p are compact and connected, then manifolds of constant p are invariant tori by Proposition 1 and the variables q are coordinates on the tori of constant p.

The dynamics on invariant tori is completely determined by a frequency which de-scribes the motion in different angular directions.

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Definition 9. Thefrequency vectorΩ is defined as Ω(p) = (Ω₁,Ω2, . . . ,Ωn)(p, q)..= ∂H0 ∂ p1, ∂H0 ∂ p2, . . . , ∂H0 ∂ pn (p, q). (2.9)

For systems given in action-angle variables the frequency only depends on p, it is con-stant on invariant tori of concon-stant p. In the definition it depends on p, q, as the notation Ω will also be used for nearly integrable systems.

Definition 10. For two degree of freedom systems withΩ1 6= 0 the ratio of the frequences is

defined as the rotational transform:

ι..= Ω2 Ω1

(2.10)

2.4 Reduction of Hamiltonian systems

If pn, qn is a pair of action-angle variables for an n degree of freedom autonomous

Hamiltonian system, then pn is an invariant and the flow is restricted to manifolds of

constant pn. This allows a local reduction of the n degree of freedom system to a family

of n−1 degree of freedom systems. In these reduced systems pn acts as a Hamiltonian,

while qn acts as a Hamiltonian time variable. This paragraph shows this reduction, it

follows the discussion on page 214 of Reference [11].

The original n degree of freedom system will be denoted by H, p, q, t, the reduced n−1 degree of freedom system by Hr, pr, qr, tr.

On each manifold{H =c0}a reduced system will be defined. An expression for Hr ..₌

pnin terms of(p1, . . . , pn−1, q1, . . . , qn−1, qn)and c0can be found by inverting

H(p1, . . . , pn, q1, . . . , qn) = c0 (2.11)

to the following expression:

pn =Hr(p1, . . . , pn−1, q1, . . . , qn−1, qn; c0) (2.12)

Under the assumption that ∂H

∂ pn 6= 0 the inverse used in Equation 2.12 exists locally, in

general a global inverse does not have to exist. By the assumption that ∂H

∂ pn 6= 0 it also follows that q

t

n(t) is strictly monotonic. This

implies that qt_n(t)can be inverted to t(qn). The following definitions give a Hamiltonian

system.

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2.5 Commutativity of subflows 17 tr ..= qn (2.13) qr .._{= (}_q₁_{, . . . , q}_n₋₁₎ _(2.14) pr .._{= (}_p₁_{, . . . , p}_n₋₁₎ _(2.15) qr,t(tr) .._{= (}_qt 1(t(tr)), . . . , qnt−1(t(tr))) (2.16) pr,t(tr) ..= (p₁t(t(tr)), . . . , pt_n₋₁(t(tr))) (2.17) Hr(pr₁, . . . , pr_n₋₁, qr₁, . . . , qr_n₋₁, tr) (2.18) Now it will be proven that Hr, pr, qr, tr as defined above satisfy Hamilton’s equations. Implicit differentiation of Equation 2.11 after filling in 2.12 for pn leads to the following:

∂H ∂q + ∂H ∂ pn ∂Hr ∂q =0 ; ∂H ∂ p + ∂H ∂ pn ∂Hr ∂ p =0 (2.19)

This gives the following equations: ∂Hr ∂ pr_i (p r_{, q}r_{, t}r_{; c}0_{) =} ∂H ∂ pi ∂H ∂ pn = dqi dt dqn dt = dq r i dtr (2.20) −∂H r ∂qr_i (p r_{, q}r_{, t}r_{; c}0_{) = −} ∂H ∂qi ∂H ∂ pn = dpi dt dqn dt = dp r i dtr (2.21)

These are exactly Hamiltons equations for the reduced system.1 The phase space of the reduced system is a(2n−2)-dimensional manifoldMr.

Note that the periodicity of H with respect to qn gives the same periodicity of Hr with

respect to tr.

Remark 2. The rotational transform ι has been defined for the two degree of freedom system (Definition 10), but it is also visible in the reduced Hamiltonian system. Therefore the term will also be used for the frequency ∂Hr

∂ pr of the reduced Hamiltonian system.

2.5 Commutativity of subflows

For a completely integrable Hamiltonian system given in action-angle variables, the system can be reduced as described in the previous paragraph. For systems with more

1_{Note that there is a minus sign mistake in the derivation, which is also there in Reference [11]. It can}

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than one degree of freedom the system can be reduced with respect to any pair pi, qi

of action-angle variables giving corresponding reduced Hamiltonian time variables tr_i. Reduction with respect to each pair of variables induces a subflow ϕtr

i =.. ϕi on the

manifold of constant pi.

It follows from the fact that the invariants pi, pjare in involution that on the intersection

of manifolds of constant pi, pj the subflows ϕi, ϕj commute ([9] page 369). This can be

seen as the reason why for completely integrable Hamiltonian systems compact and connected invariant manifolds of constant p are diffeomorphic to tori: a torus is the only compact and connected n-manifold that allows n independent commuting flows ([9] pages 368 - 369).

2.6 Poincar´e map

Each periodic Hamiltonian system has a corresponding discrete dynamical system based on the Poincar´e map. This paragraph introduces the concept of a Poincar´e map using the assumption that the Hamiltonian system is periodic in t with period T.

It needs the notion of a cross section ([11] page 214 Equation 4.8.11): for t0 ∈ R a cross

section is defined as

Σt0 _.._{= {(}_{p, q, t}_{) ∈ M}r_×R : t₌_t

0}. (2.22)

For a set U ⊂Σt0 _{the orbits of the Hamiltonian system induce the following map.}

Pt0 _{: U}_→Σt0 _(2.23)

This map is called the Poincar´e map ([11] page 214). In this thesis Poincar´e maps will be used for reduced Hamiltonian systems defined in Paragraph 2.4. The family of re-duced systems corresponding to one original Hamiltonian system is parameterised by c0, which can be displayed in notation as follows: Σt0

c0, P_ct00.

The Poincar´e map shows the time progress of the system during one period. Due to the periodicity of the Hamiltonian system the Poincar´e map can be composed with itself to study the time evolution of the system. This defines a discrete dynamical system with the cross section as phase space.

It follows from Liouville’s theorem that the discrete dynamical system defined by the Poincar´e map of a Hamiltonian system is volume preserving, which is explained in Reference [11] on pages 216 - 217. Liouville’s theorem itself follows from Hamilton’s equations.

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2.6 Poincar´e map 19

The discrete dynamical system will be used in Chapter 6 because the system is much simpler than the continuous dynamical system, but it can still be used to study the most important dynamics. A lot of information is lost in the step from the continuous to the discrete system, but the results derived for the discrete system can be interpreted in terms of the Hamiltonian system using the knowledge about how the discrete system follows from the continuous one.

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Chapter 3 Dynamical system description of a

vector field

In this chapter the behaviour of zeroes of the magnetic field and (semi-)finite field lines is shortly discussed in terms of a dynamical system. It introduces the way in which a vector field can be described by a dynamical system, also used in the next chapter. A vector field~B defined onR3with coordinates x, y, z can be identified with a dynamical system by using functions xt, yt, zt of t which solve the following autonomous system of ordinary differential equations:

dxt dt = [B· ∇x](x t_{, y}t_{, z}t₎ _(3.1) dyt dt = [B· ∇y](x t_{, y}t_{, z}t₎ _(3.2) dzt dt = [B· ∇z](x t , yt, zt) (3.3)

If xt, yt, zt satisfy these equations they parameterise field lines: the derivative of

(xt(t), yt(t), zt(t)) is a rescaled form of ~B(x, y, z). The rescaling is due to the fact that the coordinate system is not assumed to be normalised: the basis vector fields denoted by ∇x,∇y,∇z can have lengths different from 1, which allows _dtd(xt(t), yt(t), zt(t)) to be different from~B(xt(t), yt(t), zt(t)).

In terms of this dynamical system the zeroes of ~B correspond to fixed points. (Semi-) finite field lines correspond to orbits of the dynamical system converging to a zero for t → +∞ or−∞. In dynamical systems theory all points that converge to a fixed point

for t→ ∞ are the stable manifold of the fixed point. All points that converge to the fixed

point for t → −∞ are the unstable manifold. It has been shown that for smooth dynamical

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22 Dynamical system description of a vector field

All zeroes and (semi-)finite field lines of a divergenceless vector field~B can be seen as fixed points and (un)stable manifolds of the dynamical system defined by Equations 3.1 - 3.3. This gives them a smooth structure which is enough to study the relevant features of their dynamics.

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Chapter 4 A correspondence between

divergenceless vector fields and

Hamiltonian systems

The aim of this section is to show a method by which a divergenceless vector field can be described by a Hamiltonian system. The existence of a correspondence between a divergenceless vector field and a Hamiltonian system is a general property of such fields. This fact has been proven for general divergenceless vector fields in general curvilinear coordinate systems, for example in Reference [12]. That approach uses a coordinate system defined by the vector field~B. It is a useful approach to see how the divergencelessness leads to the Hamiltonian structure. The disadvantage is that it is not immediately clear whether or how replacing the Hamiltonian fuction by another function of p, q again describes a divergenceless vector field. That is required to show how results of Chapter 6 about the influence of perturbations transfer to physical vector fields. Another correspondence is described in Reference [3], where the correspondence between the vector field and the Hamiltonian system does not depend on the explicit form of the vector field, but on the the symmetry of the system, which means that it does not have the disadvantage of the method in Reference [12].

This thesis will study a more explicit construction of a Hamiltonian system than the two referred to above. Once an appropriate coordinate system has been constructed a parameterisation of field lines takes the form of Hamilton’s equations. A property of this construction is that the relation between the divergenceless vector field and the Hamiltonian system is very direct, which can be used to give conditions for the coor-dinate system that lead to a completely integrable Hamiltonian system with invariant tori in phase space. Small perturbations of such Hamiltonian systems are the subject of Chapter 6.

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24 A correspondence between divergenceless vector fields and Hamiltonian systems

Under mild conditions small perturbations of the Hamiltonian function to another func-tion of p, q give a Hamiltonian system that describes a divergenceless field as well. Therefore this method is useful to study the influence of small perturbations to an ideal system describing a vector field, it is widely used in plasma physics, for example in Reference [13].

Throughout this chapter the divergenceless vector field will be denoted by ~B. In the context of plasma physics~B is often taken to be the magnetic field, but it can also be the velocity field of an incompressible fluid.

Outline of this chapter

Paragraph 4.1 discusses the prerequisites for the coordinate system, 4.2 describes how

~_{B field lines can be parameterised and 4.3 shows that there is a Hamiltonian function}

such that the functions parameterising field lines satisfy Hamilton’s equations. This construction is applied to the Sagdeev fields in Paragraph 4.4, that procedure applies to other toroidal fields as well. Paragraph 4.5 explains how the construction can be applied to nontoroidal fields with rotational symmetries, of which field lines lie on surfaces of genus unequal to 1.

4.1 Coordinate system prerequisites

This paragraph describes properties of the coordinate system needed for the construc-tion of a Hamiltonian system in Paragraphs 4.2 - 4.3. Besides that it is described what properties of the coordinate system lead to a completely integrable Hamiltonian system with invariant tori in phase space given in action-angle variables.

Coordinate fields

In this thesis coordinate function are defined as follows.

Definition 11. A function c : Uc → Rc with Uc ⊂ R3 be an open subset and Rc a

one-dimensional manifold is a coordinate field if its derivative is nonzero on all points of Uc. The

range of c is its image, with notation: range c..₌_{im c} _⊂_R_c_.

This thesis often uses angular coordinate functions, with the following range:

range c =Rc =S1 ∼=R/2πZ (4.1)

Let c1, c2, c3 be three coordinate functions defining a coordinate system for R3, which

means that related basis vector fields ∇c1,∇c2,∇c3 are independent almost

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4.1 Coordinate system prerequisites 25

where on U ..₌ _U_c

1 ∩Uc2 ∩Uc3. The previous definitions imply that the following

map ψ is a smooth diffeomorphism when restricted to neighbourhoods of points where

∇c1,∇c2,∇c3are linearly independent.

ψ: U→range c1×range c2×range c3 : x 7→ (c1, c2, c3) (4.2)

Remark 3. In this thesis an exception is made with respect to usual definitions of coordinate systems: the map ψ does not need to be injective.

Singularities related to angular coordinate fields

The use of angular coordinates introduces singularities. A property of such coordinates is that they are not defined at the center around which the angle is defined. These singularities are allowed on curves and points of U, the coordinate transformation from new coordinates to known coordinates (x, y, z) may be noninjective on curves if the curve is a field line of~B and if its direction can be described by a basis vector field∇c. Note that the predescribed singularities are features of the basis vector fields, they are not part of the dynamical system and therefore these singularities have no influence on the application of mathematical theories to the dynamical system. However, it does change the implications of those theories for divergenceless vector fields on U.

Domain of interest

Not all field lines can be parameterised by the procedure of Paragraph 4.2. Therefore the construction of this chapter is restricted to a subset D ⊂ U ⊂ R3_{, the domain of interest.}

D must have the following properties:

• D does not contain any zeroes of~B, any (semi-)finite field lines or any singularities of the coordinate fields

• For all points in D the whole~B field line going through that point lies in D • ∇c1is defined for all points in D and is nonzero

These are the sufficient requirements for D. There is a freedom in D that can be used to exclude difficulties from calculations.

~_{B-dependent prerequisites for the coordinate system}

As Paragraph 4.2 aims to parameterise field lines of ~B, one coordinate field, c1, will

be chosen such that at all points in D the field ~B has a nonzero component in the c1

direction:

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Note that this is well defined as∇c1exists and is nonzero on D. Moreover, Appendix

A requires

~_B_{· ∇}_c₁

∇c2× ∇c3· ∇c1 (4.4)

to be a function of c2 not depending on c1, c3. The numerator is nonzero by

assump-tion 4.3 and the denominator is proporassump-tional to the determinant of the inverse Jacobian, which is nonzero. Therefore expression 4.4 is a nonzero function of c2.

This paragraph has shown that given a vector field~B onR3the field lines in a domain of interest D will be studied, where D is a union of field lines such that it does not contain any zeroes of~B or any (semi)finite field lines. On an open set U : R3 ⊃ U ⊃ D a coordinate system can be constructed. The sufficient requirements for this coordinate system needed in the rest of this chapter are summarised by the following definition. Definition 12. Apre-Hamiltonian coordinate system is a coordinate system of coordinate fields c1, c2, c3 for which expression 4.4 is a nonzero function of c2. The coordinate function

c1 is called the Hamiltonian time coordinate field. Any singularities due to angular

coor-dinate fields should be compatible with the field ~B: on D they have to be described in terms of

∇c1,∇c2,∇c3.

4.2 Field line parameterisation

Let (c1, c2, c3) be a pre-Hamiltonian coordinate system on U; D ⊂ U ⊂ R3. The

following diffeomorphism ϕ can be defined for any field line in D and any point ˆx ..= (ˆc1, ˆc2, ˆc3) ∈ D on that field line:

ϕ: R→ D : t 7→ (ct₁(t), c₂t(t), ct₃(t)); ˆc17→ ˆx (4.5)

The point ˆx has only been used to fix a point for which t = 0. The fact that it is locally diffeomorphic toR follows from 4.4, while the fact that the field line is not (semi-)finite implies that it is globally diffeomorphic. The assumption~B· ∇c1 6=0 implies that dc

t 1

dt 6=

0, which means that it is a strictly monotonic function. Together with the fact that ϕ is a diffeomorphism this means that the map ϕ can be rescaled such that

ct₁(t) = t+ˆc1 (4.6)

All field lines in D can be parameterised with base points ˆx = (0, ˆc2, ˆc3), which gives

t ≡q1.

Using the assumption that dc1t

dt = 1 it follows that dct

2

dt is the ratio of the~B component in

the∇c2 direction and the component in the∇c1direction. That ratio is proportional to

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4.3 Hamilton’s equations for divergenceless vector fields 27

the ratio of~B· ∇c2and~B· ∇c1. Proportionality follows from the fact that∇c2,∇c1may

have lengths different from 1. The same is true for dc3t

dt.

This leads to the following expressions: dc₂t dt ∝ ~_B_{· ∇}_c₂ ~_B_{· ∇}_c₁ dc₃t dt ∝ ~_B_{· ∇}_c₃ ~_B_{· ∇}_c₁

Rescaling ct₂, ct₃gives equalities similar to those in Chapter 3. dct₂ dt = ~_B_{· ∇}_c₂ ~_B_{· ∇}_c₁ (4.7) dct₃ dt = ~_B_{· ∇}_c₃ ~_B_{· ∇}_c₁ (4.8)

This paragraph has shown that under the assumptions made in Paragraph 4.1 there ex-ists a parameter t which parameterises field lines in D such that they are diffeomorphic toR. For each field line this leads to three functions ct₁(t), ct₂(t), ct₃(t)which parameterise that field line. Rescaling of these functions gives ct₁(t) =t.

4.3 Hamilton’s equations for divergenceless vector fields

This paragraph shows that there exists a Hamiltonian function such that the field line parameterisations derived in paragraph 4.2 satisfy Hamilton’s equations. It is based on the results derived in Appendix A: with ~B and a pre-Hamiltonian coordinate system

(q1, ρ, q2) a new pre-Hamiltonian coordinate system (q1, p2, q2) has been constructed

such that in these coordinates a vector potential for ~B can be written in the following form:

~

A(q1, ρ, q2) = Aq1(q1, p2, q2)∇q1+p2∇q2 (4.9)

This equation follows from A.15,A.16, leaving the tildes. For every field written in this form the function h can be defined:

h(q1, p2, q2) ..= −Aq1(q1, p2, q2) (4.10)

This definition gives the following expression for~B:

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This has effectively put all structural information about the vector field~B into the func-tion h, because all other funcfunc-tions are determined by the coordinate system.

In field lines of~B can be parameterised by functions qt₁(t) = t, p₂t(t), qt₂(t). In the rest of this paragraph it will be shown that with a function p₁t(t) that will be defined later the functions pt₁, qt₁, pt₂, qt₂satisfy Hamilton’s equations with Hamiltonian

H(p1, q1, p2, q2)..= p1+h(q1, p2, q2) (4.12)

In order to prove Hamilton’s equations a function pt₁has to be defined such that dp₁t(t)

dt = −

∂H ∂q1

(pt₁(t), qt₁(t), pt₂(t), qt₂(t))

The following function satisfies that condition: p₁t(t) ..= Z t 0 −∂h ∂q1 (qt₁(t), pt₂(t), qt₂(t))dt (4.13) Before Hamilton’s equations will be proved for the parameterisations some interme-diate results will be given. Note that the left hand side of Equation 4.14 is undefined at coordinate system singularities described in Paragraph 4.1, which is why they were excluded from D. ∇q1· ∇p2× ∇q2 6= 0 (4.14) ∇h = ∂q1h∇q1+∂p2h∇p2+∂q2h∇q2 (4.15) ~_B_{· ∇}_p₂ _{= −∇}_p₂_{· ∇}_h_{× ∇}_q₁ = −∂q2h∇q1· ∇p2× ∇q2 = −∂q2H∇q1· ∇p2× ∇q2 (4.16) ~_B_{· ∇}_q₂ ₌ _∂_p2_h_∇_q₁_{· ∇}_p₂_{× ∇}_q₂ = ∂p2H∇q1· ∇p2× ∇q2 (4.17) ~_B_{· ∇}_q₁ _{= ∇}_q₁_{· ∇}_p₂_{× ∇}_q₂ _(4.18) 28

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4.3 Hamilton’s equations for divergenceless vector fields 29

The following calculations show that the functions pt₁, qt₁, pt₂, qt₂satisfy Hamilton’s equa-tions: dpt₁ dt = − ∂H ∂q1 (4.19) dqt₁ dt = 1= ∂H ∂ p1 (4.20) dpt₂ dt = ~_B_{· ∇}_p₂ ~ B· ∇q1 = −∂q2H (4.21) dqt₂ dt = ~_B_{· ∇}_q₂ ~_B_{· ∇}_q₁ =∂p2H (4.22)

Equations 4.19,4.20 directly follow from 4.13,4.6, while 4.21,4.22 follow from 4.7,4.8 for

(c1, c2, c3) = (q1, p2, q2).

Equations 4.19 - 4.22 represent a two degree of freedom autonomous Hamiltonian sys-tem which has a four-dimensional manifoldM as phase space. M will be written as a product

M = R× M₃ (4.23)

whereR is the range of p1and

M₃ ⊂Rq1×Rp2×Rq2 (4.24)

which can be related to U by coordinate functions(c1, c2, c3) = (q1, p2, q2)and the map

ψdefined in 4.2.

Conditions under which the Hamiltonian system will be completely integrable and given in action-angle variables

The theory of Chapter 6 requires the Hamiltonian system to be completely integrable and in action-angle variables. It is useful to define the coordinate fields such that the Hamiltonian system will be in action-angle variables if that is possible, as the construc-tion of acconstruc-tion-angle coordinates is difficult (Paragraph 2.3). In order to describe~B by a Hamiltonian system in action-angle variables two of the coordinate fields q1, q2 should

be angular: range ci = Rci = S

1_{. Besides that the coordinates q}

1, q2 should commute

as follows from the discussion in Paragraph 2.5 taking in mind that the Hamiltonian time variables tr_i correspond to the variables qiof the original system, which are

coordi-nates on invariant tori in phase space. The third coordinate field p2 has to correspond

to an action variable which is an invariant, which implies that~B field lines should lie on manifolds of constant p2.

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So far this paragraph has shown that in the(q1, p2, q2)coordinate system (Appendix A)

there exists a Hamiltonian function such that the functions q₁t, pt₂, q₂t parameterising~B field lines (Paragraph 4.2) satisfy Hamilton’s equations. The Hamiltonian system will be in action-angle variables if q1, q2 are angular coordinates that commute and if field

lines lie on manifolds of constant p2.

Properties of this construction

A strength of the construction of Paragraphs 4.1 - 4.3 is that adding small perturba-tions to h under mild condiperturba-tions directly defines another divergenceless field: h is one of the components of the vector potential, replacing h gives another vector potential that corresponds to another divergenceless vector field. The mild condition is that the corresponding vector fields have to behave correctly at the boundary of the range of coordinate functions. The problem that can occur is that the new Hamiltonian system has orbits that start inside the range of the coordinate fields q1, p2, q2for qt₁(0) = ˆq1, but

has points(qt₁(t), p₂t(t), q2(t))outside the range for some values of t.

The results of this paragraph can be seen as opposite to the geometrical interpretation of dynamical systems described in Paragraph 2.2. There a Hamiltonian system is rep-resented by a vector field in phase space, while this paragraph describes a vector field inR3as a Hamiltonian system. The difference is that the vector fields in Paragraph 2.2 are defined on phase space, while this paragraph starts with a vector field onR3 and gives a Hamiltonian system, which has a corresponding vector field on phase spaceM. That vector field restricted toM3for a specific value of p1is locally diffeomorphic to the

vector field onR3on the interior of the range of the coordinate fields. It is not diffeomor-phic on the interior as the coordinate functions do not have to be injective. This fact is used in Paragraph 4.5 to describe nontoroidal vector fields by the same dynamical sys-tem as constructed for toroidal vector fields in Paragraph 4.4, which has invariant tori in phase space. The differences between the fields are incorporated in the coordinate fields, while the similarities are described by the Hamiltonian system.

4.4 A Hamiltonian system for the Sagdeev fields

In this paragraph the procedure of paragraphs 4.1 - 4.3 will be applied to the Sagdeev fields (paragraph 1.4). First the results of the procedure will be worked out in detail, while a more abstract overview of the construction of coordinate fields will be given at the end. The abstract ideas of the construction can be used for general toroidal vector fields as well as for non-toroidal vector fields by the construction in paragraph 4.5.

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4.4 A Hamiltonian system for the Sagdeev fields 31

Coordinate fields and the domain of interest

The first angular coordinate function q1is defined such that for most field lines the field

~_{B has a nonzero component in the}_∇_q₁_{direction. For the Sagdeev fields q}₁_{can be taken}

to be the angle around the z-axis. All field lines have a nonzero component in the∇q1

direction, except the field line on the z-axis, which will be excluded from D. ρ can be taken to be a specifically scaled distance to the unit circle such that surfaces of constant ρdefine the tori on which field lines of the Sagdeev fields lie.

For simplicity ∇ρ is chosen to be perpendicular to ∇q1. q2 is chosen to be the angle

around the unit circle, which is a field line of ~B and will be excluded from D. For simplicity∇q2is perpendicular to∇ρand∇q1.

A specific coordinate system that fulfills these requirements is the toroidal coordinate system described in Appendix B. Another coordinate system with differently scaled ρ, q2could have been used as well, but this coordinate system was chosen because the

Sagdeev fields have a very simple expression in these coordinates, as is shown below. For the sagdeev fields D is taken to be

D .._{= {(}_{x, y, z}_{) ∈}_R3 _{: x}2₊_y2 _>₀_{} − {(}_{x, y, z}_{) ∈} _R3 _{: x}2₊_y2 ₌_{1, z}₌₀_} _(4.25)

The Sagdeev fields have no zeroes, which implies that all field lines are bi-infinite or closed. Together with the exclusion of coordinate field singularities this implies that D as in Equation 4.25 satisfies the requirements stated in Paragraph 4.1 and that the coor-dinate system is a pre-Hamiltonian coorcoor-dinate system as needed for the construction of Appendix A.

Expressions for the vector potential and magnetic field

Computations with Wolfram Mathematica 10.3 have led to an explicit description of a vector potential for the Sagdeev fields in the toroidal coordinates of appendix B:

~ A = sech 2 ρ 2π ∇q2− ω1sech2ρ 2πω2 ∇q1 (4.26)

It follows from assumptions made between Equation 1.23 and 1.25 that ω2 6= 0, which

means that the expressions in 4.26 are well defined.

Equation 4.26 shows that for the Sagdeev fields Aρ = 0, and Aq2(ρ) = p2(ρ) does not

depend on q1, q2, which means that the vector potential is in the form of Equation 4.9.

Comparing 4.26 and 4.10 gives an expression for p2, h:

p2 = sech 2 ρ 2π (4.27) h = ω1 ω2 sech2ρ 2π = ω1 ω2p2 =ι p2 (4.28)

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Here ι is the rotational transform, which is constant. It follows that: dqt₂ dt = ∂H ∂ p2 = ∂h ∂ p2 =ι

As both q2; q1 ≡ t are 2π-periodic this implies that field lines are closed if ι = ω_ω1₂ ∈ Q

and that field lines are not closed for ω1

ω2 ∈/ Q, in that case they densely fill the invariant

tori.

As described in Appendix A p2 will be used as a coordinate field instead of ρ (Figure

4.1).

Figure 4.1:Toroidal coordinate system q1, p2, q2. Figure from [14].

The Hamiltonian system

The previous results give the following autonomous, separable two degree of freedom Hamiltonian system: H(p1, q1, p2, q2) = p1+h(p2) = p1+ω1 ω2 p2 = p1+ι p2 (4.29) dp₁t dt = 0 (4.30) dq₁t dt = 1 (4.31) dp₂t dt = 0 (4.32) dq₂t dt = ι (4.33) 32

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4.4 A Hamiltonian system for the Sagdeev fields 33

These equations imply that p1, p2, h, H are invariants defining toroidal invariant

mani-folds, where p1, h are two independent invariants as in Definition 5, just as p1, p2, both

pairs define the same invariant tori. The system is completely integrable and given in action-angle variables.

The results of this paragraph can also be stated in the reduced, one degree of freedom Hamiltonian system, which is done below. That system is simpler, which is why it will be used in Chapters 5,6.

Toroidal symmetry

The Sagdeev fields have a toroidal symmetry, the angles q1, q2 have an equivalent role.

That implies that h has a role similar to p2. If q2satisfies the prerequisites for a

Hamilto-nian time coordinate field, then the pairs h, p2and q1, q2can be mutually interchanged,

leading to a similar Hamiltonian system.

The reduced system

The Hamiltonian system of Equations 4.29 - 4.33 can be reduced as described in Para-graph 2.4. The Hamiltonian and the variables of the reduced system are related to that of the two degree of freedom system by:

tr ≡q1, qr ≡q2, pr ≡ p2, Hr ≡h

The reduced, autonomous one degree of freedom Hamiltonian system is given by the following equations. The value c0 used in the reduction is taken to be 0, as that value does not have any implications for the physical vector field it represents.

Hr(pr, qr, tr) = ω1 ω2 pr =ι pr (4.34) dHr,t dtr = 0 (4.35) dpr,t dtr = 0 (4.36) dqr,t dtr = ι (4.37)

In this system both Hr, pr are invariants (defining the same invariant manifolds), the system is completely integrable.

The construction of a coordinate system described in this paragraph can also be de-scribed in a more abstract way. In fact, it only used the existence of a core field line, the unit circle, around which other field lines spiral. q1was defined as the coordinate along

KAM and Melnikov theory describing island chains in plasmas

KAM and Melnikov theory

describing island chains in plasmas

KAM and Melnikov theory

describing island chains in plasmas

Joost Opschoor

Abstract

Introduction

Overview of this thesis

Contents

Chapter 0

Preliminary remarks

Smoothness

Chapter 1

Magnetohydrodynamics

1.1

Ideal magnetohydrodynamics (IMHD)

1.2

Resistive magnetohydrodynamics (RMHD)

1.3

Conditions used by Kamchatnov and Sagdeev

1.4

Magnetic field of the IMHD solutions studied by

Kamchatnov and Sagdeev

Chapter 2

Concepts and methods for Hamiltonian

systems

2.1

Definitions of Hamiltonian systems, separability and

integrability

∑

∑

2.2

Geometric representation of a Hamiltonian system

2.3

Action-angle variables

2.4

Reduction of Hamiltonian systems

2.5

Commutativity of subflows

2.6

Poincar´e map

Chapter 3

Dynamical system description of a

vector field

Chapter 4

A correspondence between

divergenceless vector fields and

Hamiltonian systems

4.1

Coordinate system prerequisites

4.2

Field line parameterisation

4.3

Hamilton’s equations for divergenceless vector fields

4.4

A Hamiltonian system for the Sagdeev fields